Editing Uncertainty principle (section)

==Mathematical formalism==
Starting with Kennard's derivation of position-momentum uncertainty, [[Howard Percy Robertson]] developed<ref name="Robertson1929">{{Citation|last=Robertson|first=H. P.|title=The Uncertainty Principle|journal=Phys. Rev. | year=1929|volume=34|issue=1|pages=163–164|bibcode = 1929PhRv...34..163R |doi = 10.1103/PhysRev.34.163 }}</ref><ref name=Sen2014/> a formulation for arbitrary [[Self-adjoint operator|Hermitian operator]]s <math>\hat{\mathcal{O}}</math> expressed in terms of their standard deviation
<math display="block">\sigma_{\mathcal{O}} = \sqrt{\langle \hat{\mathcal{O}}^2 \rangle-\langle \hat{\mathcal{O}}\rangle^2},</math>
where the brackets <math>\langle\hat{\mathcal{O}}\rangle</math> indicate an [[expectation value (quantum mechanics)|expectation value]] of the observable represented by operator <math>\hat{\mathcal{O}}</math>. For a pair of operators <math>\hat{A}</math> and <math>\hat{B}</math>, define their commutator as
<math display="block">[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A},</math>
and the Robertson uncertainty relation is given by<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 242–243 | bibcode = 2013qtm..book.....H }}</ref>
<math display="block">\sigma_A \sigma_B \geq \left| \frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle \right| = \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle \right|.</math>

[[Erwin Schrödinger]]<ref>Schrödinger, E., Zum Heisenbergschen Unschärfeprinzip, Berliner Berichte, 1930, pp. 296–303.</ref> showed how to allow for correlation between the operators, giving a stronger inequality, known as the '''Robertson–Schrödinger uncertainty relation''',<ref name="Schrodinger1930">{{Citation | last = Schrödinger |first = E. | title = Zum Heisenbergschen Unschärfeprinzip | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse | volume = 14 | pages = 296–303 | year = 1930}}</ref><ref name=Sen2014/>

{{Equation box 1
|indent =:
|equation = <math>\sigma_A^2\sigma_B^2 \geq \left| \frac{1}{2}\langle\{\hat{A}, \hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle \right|^2+ \left|\frac{1}{2i} \langle[ \hat{A}, \hat{B}] \rangle\right|^2,</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where the anticommutator, <math>\{\hat{A},\hat{B}\}=\hat{A}\hat{B}+\hat{B}\hat{A}</math> is used.

{{math proof
|title=Proof of the [[Erwin Schrödinger|Schrödinger]] uncertainty relation
|proof=
The derivation shown here incorporates and builds off of those shown in Robertson,<ref name="Robertson1929" /> Schrödinger<ref name="Schrodinger1930" /> and standard textbooks such as Griffiths.<ref name="GriffithsSchroeter2018">{{Cite book |last1=Griffiths |first1=David J. |url=https://www.cambridge.org/highereducation/product/9781316995433/book |title=Introduction to Quantum Mechanics |last2=Schroeter |first2=Darrell F. |year=2018 |publisher=Cambridge University Press |isbn=978-1-316-99543-3 |edition=3rd |doi=10.1017/9781316995433 |bibcode=2018iqm..book.....G |access-date=2024-01-27 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223160131/https://www.cambridge.org/highereducation/books/introduction-to-quantum-mechanics/990799CA07A83FC5312402AF6860311E#overview |url-status=live }}</ref>{{rp|138}} For any Hermitian operator <math>\hat{A}</math>, based upon the definition of [[variance]], we have
<math display="block"> \sigma_A^2 = \langle(\hat{A}-\langle \hat{A} \rangle)\Psi|(\hat{A}-\langle \hat{A} \rangle)\Psi\rangle. </math>
we let <math>|f\rangle=|(\hat{A}-\langle \hat{A} \rangle)\Psi\rangle </math> and thus
<math display="block"> \sigma_A^2 = \langle f\mid f\rangle\, .</math>

Similarly, for any other Hermitian operator <math> \hat{B} </math> in the same state
<math display="block"> \sigma_B^2 = \langle(\hat{B}-\langle \hat{B} \rangle)\Psi|(\hat{B}-\langle \hat{B} \rangle)\Psi\rangle = \langle g\mid g\rangle </math>
for <math> |g\rangle=|(\hat{B}-\langle \hat{B} \rangle)\Psi \rangle.</math>

The product of the two deviations can thus be expressed as

{{NumBlk|:|<math> \sigma_A^2\sigma_B^2 = \langle f\mid f\rangle\langle g\mid g\rangle. </math>|{{EquationRef|1}}}}

In order to relate the two vectors <math>|f\rangle</math> and <math>|g\rangle</math>, we use the [[Cauchy–Schwarz inequality]]<ref name="Riley2006">{{Citation | last = Riley | first = K. F. | author2 = M. P. Hobson and S. J. Bence | title = Mathematical Methods for Physics and Engineering | publisher = Cambridge | year = 2006 | page = 246 }}{{ISBN?}}</ref> which is defined as
<math display="block">\langle f\mid f\rangle\langle g\mid g\rangle \geq |\langle f\mid g\rangle|^2, </math>
and thus Equation ({{EquationNote|1}}) can be written as

{{NumBlk|:|<math>\sigma_A^2\sigma_B^2 \geq |\langle f\mid g\rangle|^2.</math>|{{EquationRef|2}}}}

Since <math> \langle f\mid g\rangle</math> is in general a complex number, we use the fact that the modulus squared of any complex number <math>z</math> is defined as <math>|z|^2=zz^{*}</math>, where <math>z^{*}</math> is the complex conjugate of <math>z</math>. The modulus squared can also be expressed as

{{NumBlk|:|<math> |z|^2 = \Big(\operatorname{Re}(z)\Big)^2+\Big(\operatorname{Im}(z)\Big)^2 = \Big(\frac{z+z^\ast}{2}\Big)^2 +\Big(\frac{z-z^\ast}{2i}\Big)^2. </math>|{{EquationRef|3}}}}

we let <math>z=\langle f\mid g\rangle</math> and <math>z^{*}=\langle g \mid f \rangle </math> and substitute these into the equation above to get

{{NumBlk|:|<math>|\langle f\mid g\rangle|^2 = \bigg(\frac{\langle f\mid g\rangle+\langle g\mid f\rangle}{2}\bigg)^2 + \bigg(\frac{\langle f\mid g\rangle-\langle g\mid f\rangle}{2i}\bigg)^2 </math>|{{EquationRef|4}}}}

The inner product <math>\langle f\mid g\rangle </math> is written out explicitly as
<math display="block">\langle f\mid g\rangle = \langle(\hat{A}-\langle \hat{A} \rangle)\Psi|(\hat{B}-\langle \hat{B} \rangle)\Psi\rangle,</math>
and using the fact that <math>\hat{A}</math> and <math>\hat{B}</math> are Hermitian operators, we find
<math display="block">
\begin{align}
\langle f\mid g\rangle & = \langle\Psi|(\hat{A}-\langle \hat{A}\rangle)(\hat{B}-\langle \hat{B}\rangle)\Psi\rangle \\[4pt]
& = \langle\Psi\mid(\hat{A}\hat{B}-\hat{A}\langle \hat{B}\rangle - \hat{B}\langle \hat{A}\rangle + \langle \hat{A}\rangle\langle \hat{B}\rangle)\Psi\rangle \\[4pt]
& = \langle\Psi\mid\hat{A}\hat{B}\Psi\rangle-\langle\Psi\mid\hat{A}\langle \hat{B}\rangle\Psi\rangle
-\langle\Psi\mid\hat{B}\langle \hat{A}\rangle\Psi\rangle+\langle\Psi\mid\langle \hat{A}\rangle\langle \hat{B}\rangle\Psi\rangle \\[4pt]
& =\langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle+\langle \hat{A}\rangle\langle \hat{B}\rangle \\[4pt]
& =\langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle.
\end{align}
</math>

Similarly it can be shown that <math>\langle g\mid f\rangle = \langle \hat{B}\hat{A}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle.</math>

Thus, we have
<math display="block">
\langle f\mid g\rangle-\langle g\mid f\rangle = \langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle-\langle \hat{B}\hat{A}\rangle+\langle \hat{A}\rangle\langle \hat{B}\rangle = \langle [\hat{A},\hat{B}]\rangle
</math>
and
<math display="block">\langle f\mid g\rangle+\langle g\mid f\rangle = \langle \hat{A}\hat{B}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle+\langle \hat{B}\hat{A}\rangle-\langle \hat{A}\rangle\langle \hat{B}\rangle = \langle \{\hat{A},\hat{B}\}\rangle -2\langle \hat{A}\rangle\langle \hat{B}\rangle. </math>

We now substitute the above two equations above back into Eq. ({{EquationNote|4}}) and get
<math display="block">|\langle f\mid g\rangle|^2=\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle\Big)^2 + \Big(\frac{1}{2i} \langle[\hat{A},\hat{B}]\rangle\Big)^{2}\, .</math>

Substituting the above into Equation ({{EquationNote|2}}) we get the Schrödinger uncertainty relation
<math display="block">\sigma_A\sigma_B \geq \sqrt{\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}\rangle - \langle \hat{A} \rangle\langle \hat{B}\rangle\Big)^2 + \Big(\frac{1}{2i} \langle[\hat{A},\hat{B}]\rangle\Big)^2}.</math>

This proof has an issue<ref>{{Citation|last=Davidson|first=E. R.|title=On Derivations of the Uncertainty Principle|journal=J. Chem. Phys.|volume=42|year=1965|doi=10.1063/1.1696139|bibcode = 1965JChPh..42.1461D|issue=4|pages=1461–1462 }}</ref> related to the domains of the operators involved. For the proof to make sense, the vector <math> \hat{B} |\Psi \rangle</math> has to be in the domain of the [[unbounded operator]] <math> \hat{A}</math>, which is not always the case. In fact, the Robertson uncertainty relation is false if <math>\hat{A}</math> is an angle variable and <math>\hat{B}</math> is the derivative with respect to this variable. In this example, the commutator is a nonzero constant—just as in the Heisenberg uncertainty relation—and yet there are states where the product of the uncertainties is zero.<ref name="Hall2013"/> (See the counterexample section below.) This issue can be overcome by using a [[variational method]] for the proof,<ref name="Jackiw">{{Citation|last=Jackiw| first=Roman|title=Minimum Uncertainty Product, Number-Phase Uncertainty Product, and Coherent States|journal=J. Math. Phys.|volume=9|year=1968|doi=10.1063/1.1664585|bibcode = 1968JMP.....9..339J|issue=3|pages=339–346 }}</ref><ref name="CarruthersNieto">{{Citation|first1=P. |last1=Carruthers|last2= Nieto|first2=M. M.|title=Phase and Angle Variables in Quantum Mechanics|journal=Rev. Mod. Phys.|volume=40|year=1968|doi=10.1103/RevModPhys.40.411|bibcode = 1968RvMP...40..411C|issue=2|pages=411–440 }}</ref> or by working with an exponentiated version of the canonical commutation relations.<ref name="Hall2013"/>

Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators <math>\hat{A}</math> and <math>\hat{B}</math> are [[Self-adjoint operator#Self-adjoint operators|self-adjoint operators]]. It suffices to assume that they are merely [[Self-adjoint operator#Symmetric operators|symmetric operators]]. (The distinction between these two notions is generally glossed over in the physics literature, where the term ''Hermitian'' is used for either or both classes of operators. See Chapter 9 of Hall's book<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | bibcode = 2013qtm..book.....H }}</ref> for a detailed discussion of this important but technical distinction.)
}}

===Phase space===
In the [[phase space formulation]] of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a [[Wigner quasi-probability distribution|Wigner function]] <math>W(x,p)</math> with [[Moyal product|star product]] ★ and a function ''f'', the following is generally true:<ref>{{Cite journal | last1 = Curtright | first1 = T. |last2= Zachos | first2= C. | title = Negative Probability and Uncertainty Relations| journal = Modern Physics Letters A | volume = 16 | issue = 37 | pages = 2381–2385 | doi = 10.1142/S021773230100576X | year = 2001 |arxiv = hep-th/0105226 |bibcode = 2001MPLA...16.2381C | s2cid = 119669313 }}</ref>
<math display="block">\langle f^* \star f \rangle =\int (f^* \star f) \, W(x,p) \, dx \, dp \ge 0 ~.</math>

Choosing <math>f = a + bx + cp</math>, we arrive at
<math display="block">\langle f^* \star f \rangle =\begin{bmatrix}a^* & b^* & c^* \end{bmatrix}\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix} \ge 0 ~.</math>

Since this positivity condition is true for ''all'' ''a'', ''b'', and ''c'', it follows that all the eigenvalues of the matrix are non-negative.

The non-negative eigenvalues then imply a corresponding non-negativity condition on the [[determinant]],
<math display="block">\det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}
= \det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x^2 \rangle & \left\langle xp + \frac{i\hbar}{2} \right\rangle \\ \langle p \rangle & \left\langle xp - \frac{i\hbar}{2} \right\rangle & \langle p^2 \rangle \end{bmatrix}
\ge 0~,</math>
or, explicitly, after algebraic manipulation,
<math display="block">\sigma_x^2 \sigma_p^2 = \left( \langle x^2 \rangle - \langle x \rangle^2 \right)\left( \langle p^2 \rangle - \langle p \rangle^2 \right)\ge \left( \langle xp \rangle - \langle x \rangle \langle p \rangle \right)^2 + \frac{\hbar^2}{4} ~.</math>

===Examples===

Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below.
* '''Position–linear momentum uncertainty relation''': for the position and linear momentum operators, the canonical commutation relation <math>[\hat{x}, \hat{p}] = i\hbar</math> implies the Kennard inequality from above: <math display="block">\sigma_x \sigma_p \geq \frac{\hbar}{2}.</math>
* '''Angular momentum uncertainty relation''': For two orthogonal components of the [[angular momentum|total angular momentum]] operator of an object: <math display="block">\sigma_{J_i} \sigma_{J_j} \geq \frac{\hbar}{2} \big|\langle J_k\rangle\big|,</math> where ''i'', ''j'', ''k'' are distinct, and ''J''<sub>''i''</sub> denotes angular momentum along the ''x''<sub>''i''</sub> axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for <math>[J_x, J_y] = i \hbar \varepsilon_{xyz} J_z</math>, a choice <math>\hat{A} = J_x</math>, <math>\hat{B} = J_y</math>, in angular momentum multiplets, ''ψ'' = |''j'', ''m''⟩, bounds the [[Casimir invariant]] (angular momentum squared, <math>\langle J_x^2+ J_y^2 + J_z^2 \rangle</math>) from below and thus yields useful constraints such as {{nobr|''j''(''j'' + 1) ≥ ''m''(''m'' + 1)}}, and hence ''j'' ≥ ''m'', among others.

* For the number of electrons in a [[superconductor]] and the [[Phase factor|phase]] of its [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]]<ref>{{Citation |last=Likharev |first=K. K. |author2=A. B. Zorin |title=Theory of Bloch-Wave Oscillations in Small Josephson Junctions |journal=J. Low Temp. Phys. |volume=59 |issue=3/4 |pages=347–382 |year=1985 |doi=10.1007/BF00683782 |bibcode=1985JLTP...59..347L|s2cid=120813342 }}</ref><ref>{{Citation |first=P. W. |last=Anderson |editor-last=Caianiello |editor-first=E. R. |contribution=Special Effects in Superconductivity |title=Lectures on the Many-Body Problem, Vol. 2 |year=1964 |place=New York |publisher=Academic Press}}</ref> <math display="block"> \Delta N \, \Delta \varphi \geq 1. </math>

===Limitations===
The derivation of the Robertson inequality for operators <math>\hat{A}</math> and <math>\hat{B}</math> requires <math>\hat{A}\hat{B}\psi</math> and <math>\hat{B}\hat{A}\psi</math> to be defined. There are quantum systems where these conditions are not valid.<ref>{{Cite journal |last=Davidson |first=Ernest R. |date=1965-02-15 |title=On Derivations of the Uncertainty Principle |url=https://pubs.aip.org/jcp/article/42/4/1461/208937/On-Derivations-of-the-Uncertainty-Principle |journal=The Journal of Chemical Physics |language=en |volume=42 |issue=4 |pages=1461–1462 |doi=10.1063/1.1696139 |bibcode=1965JChPh..42.1461D |issn=0021-9606 |access-date=2024-01-20 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223160247/https://pubs.aip.org/aip/jcp/article-abstract/42/4/1461/208937/On-Derivations-of-the-Uncertainty-Principle?redirectedFrom=fulltext |url-status=live }}</ref>
One example is a quantum [[particle in a ring|particle on a ring]], where the wave function depends on an angular variable <math>\theta</math> in the interval <math>[0,2\pi]</math>. Define "position" and "momentum" operators <math>\hat{A}</math> and <math>\hat{B}</math> by
<math display="block">\hat{A}\psi(\theta)=\theta\psi(\theta),\quad \theta\in [0,2\pi],</math>
and
<math display="block">\hat{B}\psi=-i\hbar\frac{d\psi}{d\theta},</math>
with periodic boundary conditions on <math>\hat{B}</math>. The definition of <math>\hat{A}</math> depends the <math>\theta</math> range from 0 to <math>2\pi</math>. These operators satisfy the usual commutation relations for position and momentum operators, <math>[\hat{A},\hat{B}]=i\hbar</math>. More precisely, <math>\hat{A}\hat{B}\psi-\hat{B}\hat{A}\psi=i\hbar\psi</math> whenever both <math>\hat{A}\hat{B}\psi</math> and <math>\hat{B}\hat{A}\psi</math> are defined, and the space of such <math>\psi</math> is a dense subspace of the quantum Hilbert space.<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | page = 245 | bibcode = 2013qtm..book.....H }}</ref>

Now let <math>\psi</math> be any of the eigenstates of <math>\hat{B}</math>, which are given by <math>\psi(\theta)=e^{2\pi in\theta}</math>. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator <math>\hat{A}</math> is bounded, since <math>\theta</math> ranges over a bounded interval. Thus, in the state <math>\psi</math>, the uncertainty of <math>B</math> is zero and the uncertainty of <math>A</math> is finite, so that
<math display="block">\sigma_A\sigma_B=0.</math>
The Robertson uncertainty principle does not apply in this case: <math>\psi</math> is not in the domain of the operator <math>\hat{B}\hat{A}</math>, since multiplication by <math>\theta</math> disrupts the periodic boundary conditions imposed on <math>\hat{B}</math>.<ref name="Hall2013">{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 245 | bibcode = 2013qtm..book.....H }}</ref>

For the usual position and momentum operators <math>\hat{X}</math> and <math>\hat{P}</math> on the real line, no such counterexamples can occur. As long as <math>\sigma_x</math> and <math>\sigma_p</math> are defined in the state <math>\psi</math>, the Heisenberg uncertainty principle holds, even if <math>\psi</math> fails to be in the domain of <math>\hat{X}\hat{P}</math> or of <math>\hat{P}\hat{X}</math>.<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 246 | bibcode = 2013qtm..book.....H }}</ref>

===Mixed states===

The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components <math>\varrho_k</math> in any decomposition of the [[density matrix]] given as
<math display="block">
\varrho=\sum_k p_k \varrho_k.
</math>
Here, for the probabilities <math>p_k\ge0</math> and <math>\sum_k p_k=1</math> hold. Then, using the relation
<math display="block">
\sum_k a_k \sum_k b_k \ge \left(\sum_k \sqrt{a_k b_k}\right)^2
</math>
for <math> a_k,b_k\ge 0</math>,
it follows that<ref name="PhysRevResearch21">{{cite journal |last1=Tóth |first1=Géza |last2=Fröwis |first2=Florian |title=Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013075 |doi=10.1103/PhysRevResearch.4.013075|arxiv=2109.06893 |bibcode=2022PhRvR...4a3075T |s2cid=237513549 }}</ref>
<math display="block">
\sigma_A^2 \sigma_B^2 \geq \left[\sum_k p_k L(\varrho_k)\right]^2,
</math>
where the function in the bound is defined
<math display="block">
L(\varrho) = \sqrt{\left | \frac{1}{2}\operatorname{tr}(\rho\{A,B\}) - \operatorname{tr}(\rho A)\operatorname{tr}(\rho B)\right |^2 +\left | \frac{1}{2i} \operatorname{tr}(\rho[A,B])\right | ^2}.
</math>
The above relation very often has a bound larger than that of the original Robertson–Schrödinger uncertainty relation. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. The following expression is stronger than the Robertson–Schrödinger uncertainty relation
<math display="block">
\sigma_A^2 \sigma_B^2 \geq \left[\max_{p_k,\varrho_k} \sum_k p_k L(\varrho_k)\right]^2,
</math>
where on the right-hand side there is a concave roof over the decompositions of the density matrix.
The improved relation above is saturated by all single-qubit quantum states.<ref name="PhysRevResearch21" />

With similar arguments, one can derive a relation with a convex roof on the right-hand side<ref name="PhysRevResearch21" />
<math display="block">
\sigma_A^2 F_Q[\varrho,B] \geq 4 \left[\min_{p_k,\Psi_k} \sum_k p_k L(\vert \Psi_k\rangle\langle \Psi_k\vert)\right]^2
</math>
where <math>F_Q[\varrho,B]</math> denotes the [[quantum Fisher information]] and the density matrix is decomposed to pure states as
<math display="block">
\varrho=\sum_k p_k \vert \Psi_k\rangle \langle \Psi_k\vert.
 </math>
The derivation takes advantage of the fact that the [[quantum Fisher information]] is the convex roof of the variance times four.<ref>{{cite journal |last1=Tóth |first1=Géza |last2=Petz |first2=Dénes |title=Extremal properties of the variance and the quantum Fisher information |journal=Physical Review A |date=20 March 2013 |volume=87 |issue=3 |pages=032324 |doi=10.1103/PhysRevA.87.032324|bibcode=2013PhRvA..87c2324T |arxiv=1109.2831 |s2cid=55088553 }}</ref><ref>{{cite arXiv |last1=Yu |first1=Sixia |title=Quantum Fisher Information as the Convex Roof of Variance |date=2013 |eprint=1302.5311|class=quant-ph }}</ref>

A simpler inequality follows without a convex roof<ref>{{cite journal |last1=Fröwis |first1=Florian |last2=Schmied |first2=Roman |last3=Gisin |first3=Nicolas |title=Tighter quantum uncertainty relations following from a general probabilistic bound |journal=Physical Review A |date=2 July 2015 |volume=92 |issue=1 |pages=012102 |doi=10.1103/PhysRevA.92.012102|arxiv=1409.4440 |bibcode=2015PhRvA..92a2102F |s2cid=58912643 }}</ref>
<math display="block">
\sigma_A^2 F_Q[\varrho,B] \geq \vert \langle i[A,B]\rangle\vert^2,
</math>
which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we have
<math display="block">
F_Q[\varrho,B]\le 4 \sigma_B,
</math>
while for pure states the equality holds.

===The Maccone–Pati uncertainty relations===
The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Lorenzo Maccone and [[Arun K. Pati]] give non-trivial bounds on the sum of the variances for two incompatible observables.<ref>{{cite journal|last1=Maccone|first1=Lorenzo|last2=Pati|first2=Arun K.|title=Stronger Uncertainty Relations for All Incompatible Observables|journal=Physical Review Letters|date=31 December 2014|volume=113| issue=26|page=260401|doi=10.1103/PhysRevLett.113.260401|pmid=25615288|arxiv=1407.0338|bibcode=2014PhRvL.113z0401M|s2cid=21334130 }}</ref> (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref.<ref>{{cite journal |last1=Huang |first1=Yichen |title=Variance-based uncertainty relations |journal=Physical Review A |date=10 August 2012 |volume=86 |issue=2 |page=024101 |doi=10.1103/PhysRevA.86.024101|arxiv=1012.3105 |bibcode=2012PhRvA..86b4101H |s2cid=118507388 }}</ref> due to Yichen Huang.) For two non-commuting observables <math>A</math> and <math>B</math> the first stronger uncertainty relation is given by
<math display="block"> \sigma_{A}^2 + \sigma_{ B}^2 \ge \pm i \langle \Psi\mid [A, B]|\Psi \rangle + \mid \langle \Psi\mid(A \pm i B)\mid{\bar \Psi} \rangle|^2, </math>
where <math> \sigma_{A}^2 = \langle \Psi |A^2 |\Psi \rangle - \langle \Psi \mid A \mid \Psi \rangle^2 </math>, <math> \sigma_{B}^2 = \langle \Psi |B^2 |\Psi \rangle - \langle \Psi \mid B \mid\Psi \rangle^2 </math>, <math>|{\bar \Psi} \rangle </math> is a normalized vector that is orthogonal to the state of the system <math>|\Psi \rangle </math> and one should choose the sign of <math>\pm i \langle \Psi\mid[A, B]\mid\Psi \rangle </math> to make this real quantity a positive number.

The second stronger uncertainty relation is given by
<math display="block"> \sigma_A^2 + \sigma_B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} \mid(A + B)\mid \Psi \rangle|^2 </math>
where <math>| {\bar \Psi}_{A+B} \rangle </math> is a state orthogonal to <math> |\Psi \rangle </math>.
The form of <math>| {\bar \Psi}_{A+B} \rangle </math> implies that the right-hand side of the new uncertainty relation is nonzero unless <math>| \Psi\rangle </math> is an eigenstate of <math>(A + B)</math>. One may note that <math>|\Psi \rangle </math> can be an eigenstate of <math>( A+ B)</math> without being an eigenstate of either <math> A</math> or <math> B </math>. However, when <math> |\Psi \rangle </math> is an eigenstate of one of the two observables the Heisenberg–Schrödinger uncertainty relation becomes trivial. But the lower bound in the new relation is nonzero unless <math> |\Psi \rangle </math> is an eigenstate of both.